Other Triadic Tunings

The notion of a triadic tuning can be useful to resolve certain theoretical dilemmas that arise in diatonic tunings in which a d4 is tuned such that it forms a better M3 than M3 itself. For example, in Pythagorean tuning, M3 are $C_S \approx 17.9$ mil away from just, but d4 () are only $C_S - C_P \approx -1.6$ mil away from a just M3. Thus it is desirable to render a notated M3 like C3--E3 as C3--F, although at other times the ``real'' E3 would be desirable, for instance in the P5 A3--E3. The solution to this dilemma is to construct a Pythagorean triadic tuning

$$\ensuremath{\ensuremath{\underline{\tau}}_{{}}\!\left(\mbox{... ...ht)x_t + \ensuremath{\underline{3/2}}\:x\hspace{-1pt}_f+ x_r\:.$$

Reformulating the situation in this way shows that this is simply an instance of triadic tuning, not some aberrant use of Pythagorean diatonic tuning.

This reformulation is also useful in understanding the approximation to just intonation offered by large equal temperaments such as 53TET, which have attracted theoretical attention since the early 17th century [14, 148]. The triadic formulation of 53TET is

$$\ensuremath{\ensuremath{\underline{\tau}}_{{}}\!\left(\mbox{\boldmath$x$}\right)} = (17/53)x_t + (31/53)x\hspace{-1pt}_f+ x_r.$$

As one might imagine, the problems of P1₁ slips still apply to these other triadic tunings.